Pattern Recognition and Tomographic Reconstruction with Terahertz Signals for Applications in Biomedical Engineering by Xiaoxia (Sunny) Yin Bachelor of Engineering (Industrial Electronics), Dalian University, 1996 Thesis submitted for the degree of Doctor of Philosophy in Electrical and Electronic Engineering University of Adelaide 2008 Appendix A Oblique Projection Operation HIS appendix introduces an oblique projection operation, which T underpins the subspace system identification algorithms dis- cussed in Chapter 7. In addition, a few figures are drawn to il- lustrate parts of the subspace identification procedures. Page 289 U Y M Y/U M Figure A.1. Schematic of oblique projection. Oblique projection of Y onto M along the direction of U. An oblique projection operation is illustrated in Fig. A.1. In this case, there are three rows spaces involved: Y, U and M. The notation Y/UM is used to denote the oblique projection of Y onto M along the direction of U. The vector space Y in the oblique projection should lie in the span of U, M , otherwise it should be first projected or- { } thogonally onto that span before the oblique projection operation is carried out. The oblique projection is defined by two properties, which are obvious from Fig. A.1: U/UM = 0 (A.1) M/UM = M. (A.2) Any transformation that satisfies Eq. (A.1) and Eq. (A.2) can be adopted as an oblique projection operator. The following formula satisfies the conditions for the projection operator: M/UM = (Y/U⊥)(M/U⊥)∗ M (A.3) where denotes the Moore-Penrose pseudo-inverse, indicates an orthogonal projec- ∗ ⊥ tion. In fact, Eq. (A.1) follows from the fact that (U/U⊥) is zero by definition, whereas Eq. (A.3) is satisfied because (M/U⊥)(M/U⊥)∗ equals the identity matrix. Oblique projection vectors are also drawn in Fig. A.2(a) to represent the row spaces of the involved matrices in Eq. (7.20), Section 7.5.1. The row space Y f is seen to be the sum Page 290 Appendix A Oblique Projection Operation - - Ωi Uf Y 1 Y f Ωi- Uf f (a) (b) + = Γ X W O -1 = Γ X 1 W Oi i f p i i i+ p Figure A.2. The projections regarding state-space sequences. Determination of (a) the future et al. state sequence X f and (b) the shifted state sequence Xi+1. After Galv˜ao (2005). Γ Ω Ω of iX f and iU f . In this projection operation, the direction of iU f is known, since Ω Ω U f , consisting of input data, is known, and i, albeit unknown, iU f can be realised via the performance of rescalings and rotations of the rows of U f in the hyperplane in which they lie. As a consequence, the oblique projection of Y f along U f or along Ω iU f is the same. The direction of Wp is also known, composed of input and output data. Moreover, if the system is observable, it is easy to see that X f lies in the Wp space, because x[i] can be estimated according to the input u[k] and output y[k] data up to the instant k = i 1. As a result, we can obtain the direction with respect to the − oblique projection of Y f . Owing to the availability of known deterministic input data, the identification allows to be carried out by performing an oblique projection. The augmented matrix A¯ has a column zeros, which results in an eigenvalue at z = 0. 1 In fact, an eigenvalue at the origin corresponds to the z− delay factor, which intro- duces a pole at z = 0 in the transfer function G¯ d(z). Fig. A.2(b) illustrates the projection procedure regarding the shifted state sequence, which is represented in Eq. (7.28), Section 7.5.1. It is similar to the reasoning related to Fig. A.2(a). The Oi 1 is the oblique projection of Y− (with omitting of the first row − f of the matrix) along Hi 1U− direction. It must be performed on the row space of the − f + expanded matrix of input-output data Wp , which is obtained from Wp by adding one row at the bottom. Page 291 Page 292 Appendix B Back Projection Algorithms HIS Appendix provides further details about back projection al- T gorithms. This is a specific supplement made for Chapter 10 in respect of computed tomography reconstruction. Page 293 B.1 Theory B.1 Theory The back projection is represented via parallel beam projections. Recalling the formula for the inverse Fourier transform, the object function, f (x, y), can be expressed as ∞ ∞ f (x, y) = F(u, v)ej2π(ux+vy)dudv. (B.1) ∞ ∞ Z− Z− Exchanging the rectangular coordinate system in the frequency domain, (u, v), for a polar coordinate system, (w, θ), by making the substitutions u = w cos θ (B.2) v = w sin θ (B.3) and changing the differentials by using dudv = wdwdθ, (B.4) we can write the inverse Fourier transform of a polar function as 2π ∞ f (x, y) = F(w, θ)ej2πw(x cos θ+y sin θ)wdwdθ. (B.5) Z0 Z0 This integral can be split into two by considering θ from 0◦ to 180◦ and then from 180◦ to 360◦, π ∞ f (x, y) = F(w, θ)ej2πw(x cos θ+y sin θ)wdwdθ 0 0 Z Zπ ∞ + F(w, θ)ej2πw[x cos(θ+180◦)+y sin(θ+180◦)]wdwdθ (B.6) Z0 Z0 and then using the property F(w, θ + 180◦) = F( w, θ) (B.7) − the above expression for f (x, y) may be written as π ∞ f (x, y) = F(w, θ) w ej2πwtdw dθ. (B.8) 0 ∞ | | Z Z− Here, we have simplified the expression by setting t = x cos θ + y sin θ. (B.9) If we substitute the Fourier transform of the projection at angle θ, sθ(w), for the two- dimensioinal Fourier transform F(w, θ), we get π ∞ j2πwt f (x, y) = sθ(w) w e dw dθ. (B.10) 0 ∞ | | Z Z− Page 294 Appendix B Back Projection Algorithms This integral in Eq. (B.10) may be expressed as π f (x, y) = Qθ( x cos θ + y sin θ)dθ (B.11) Z0 where ∞ j2πwt Qθ = Sθ (w) w e dw. (B.12) ∞ | | Z− Eq. (B.11) represents a filtering operation, where the frequency response of the filter is given by w ; therefore Q (w) is called a ‘filtered projection’. The resulting projections | | θ for different angles θ are then back projected to form the estimate of f (x, y). We relabel Sθ(w) to S(θ, β),ot t ξ, then we rewrite Eq. (B.11) to yield, π ∞ I(x, y) = S(θ, β) β exp[i 2πβξ]dβ dθ (B.13) 0 ∞ | | Z "Z− # where, Eq. (B.13) is the same as Eq. (10.2) that we use in Section 10.2 for THz recon- struction. Page 295 Page 296 Appendix C Error Analysis Regarding Wavelet Based Local Reconstruction HIS Appendix provides further details about error analysis with T respect to wavelet based local reconstruction. This is a specific supplement made for Chapter 12 to validate local CT via wavelet transforms. Page 297 C.1 Methodology C.1 Methodology Radon transform error is not negligible because of the nonlocal property of the deriva- tive Hilbert transform (the impulse response of the filter β ). In this case, even a small | | local ROI can be reconstructed by considering some data outside the region of interest for a negligible reconstruction error. In terms of the amount of nonlocal data applied in the reconstruction, an upper bound of the reconstruction error can be calculated. The comparison is made between the wavelet based reconstruction and the traditional reconstruction algorithm for the local tomography image recovery. In the current algorithm, the region of interest (ROI) and the region of exposure (ROE), are assumed to be centered at the center of an image. The support of a completed image is a disk of radius R pixels centered at the origin. Disks of radius ri pixels and re pixels centered at the origin are used to denote the ROI and ROE, respectively. Consider the Eq. (C.5), the traditional filtered back projection algorithm, which is shifted to the time domain scheme. π Ir(x, y) = s(θ, ξ)hθ ( x cos θ + y sin θ)dθ. (C.1) Z0 The reconstructed function Ir(x, y) is an approximation of the function I(x, y) ifs hθ i the angle dependent impulse response of the ramp filter, θ [0, 2π), and is an ap- ∈ proximation of the wavelet and scaling coefficients if wavelet and scaling filters are substituted for β . | | The discrete version is expressed as follows: π k 1 R Ir(x, y) = ∑ ∑ sθk (n)hθk (m n) (C.2) k k=1 R n= R − − where m = ( x cos θ + y sin θ) ROE, K is the number of the measured projection ∈ angles, sθk indicates the projection at kth angle and θk = k(π/k). The completed image based on global data consists of two parts: the ROE and its complement ROE, π k 1 π k 1 Ir(x, y) = ∑ ∑ s (n)h (m n) + ∑ ∑ s (n)h (m n). (C.3) k R θk θk k R θk θk k=1 n r − k=1 n >r − |≤| e | | e Therefore the magnitude of error regarding the ROE can be calculated as follows: π k 1 ǫ(x, y) = ∑ ∑ sθ (n)hθ (m n) .